The twodimensional nature of polariton condensates implies that the
phase transition between the normal and superfluid state at
equilibrium is of the BKT type. Numerical simulations have provided
evidence that a similar transition exists out of equilibrium
[1]}, [2]}, [3]}, [4]}, even though
studies based on the renormalization group have pointed out that
the phase transition is crucially affected at long distances by the
Kardar-Parisi-Zhang dynamics of the phase, that tends to destroy the
ordered phase [5]}, [6]}, [7]}, [8]}. In practice,
however, the KPZ physics can be limited to very large system sizes,
so that in experimental 2D systems the BKT-like physics dominates
[1]}, [2]}, [3]}, [4]}. Recently, we have
shown by analytical arguments and numerical simulations that also
photon condensates without interactions do feature a BKT like phase
transition, that is stabilized by the driving and
dissipation [13]}. The question of the dynamical
formation of the ordered state after a rapid switching of the
pumping intensity is therefore also relevant for photon condensates.
